Finite difference schemes for k ⋅ p models: A comparative study

Multi-band k · p models discretized with finite difference method (FDM) have been widely used to study electronic properties of semiconductor nanostructures. However, different schemes of FDM exist in the literature, some of them are numerically unstable leading to spurious states [1][2], while others are stable but require special treatment of the boundary conditions and/or the material interfaces [3][4][5][6]. Therefore, a comparison of their numerical behaviors (and implementation tricks) will be very helpful for selecting a suitable scheme and obtaining reliable results. To this end, we have implemented into NEMO5 simulation software [7] the following options, (a) centered difference for symmetrized (SYM) Hamiltonian [1], (b) centered difference for Burt-Foreman (BF) Hamiltonian [8], (c) one-sided differences for SYM Hamiltonian [3], and (d) one-sided differences for BF Hamiltonian [6]. For all cases, eight-band and six-band models for both zincblende and wurtzite type materials are available.